Analyses of the cosmic microwave background (CMB) radiationmaps made by the Wilkinson Microwave Anisotropy Probe (WMAP) have revealed anomalies not predicted by the standard inflationary cosmology. In particular, the power of the quadrupole moment of the CMB fluctuations is remarkably low, and the quadrupole and octopole moments are aligned mutually and with the geometry of the solar system. It has been suggested in the literature that microwave sky pollution by an unidentified dust cloud in the vicinity of the solar system may be the cause for these anomalies. In this paper, we simulate the thermal emission by clouds of spherical homogeneous particles of several materials. Spectral constraints from the WMAP multi-wavelength data and earlier infrared observations on the hypothetical dust cloud are used to determine the dust cloud's physical characteristics. In order for its emissivity to demonstrate a flat, CMB-like wavelength dependence over the WMAP wavelengths ( 3 through 14 mm), and to be invisible in the infrared light, its particles must be macroscopic. Silicate spheres of several millimeters in size and carbonaceous particles an order of magnitude smaller will suffice. According to our estimates of the abundance of such particles in the zodiacal cloud and trans-Neptunian belt, yielding the optical depths of the order of 10(-7) for each cloud, the solar system dust can well contribute 10 mu K ( within an order of magnitude) in the microwaves. This is not only intriguingly close to the magnitude of the anomalies ( about 30 mu K), but also alarmingly above the presently believed magnitude of systematic biases of the WMAP results ( below 5 mu K) and, to an even greater degree, of the future missions with higher sensitivities, e. g., Planck.